##### Department of Mathematics,

University of California San Diego

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### Math 209: Number Theory Seminar

## Aranya Lahiri

#### UC San Diego

## Dagger groups and $p$-adic distribution algebras

##### Abstract:

Locally analytic representations were introduced by Peter Schneider and Jeremy Teitelbaum as a tool to understand $p$-adic Langlands program. From the very beginning the theory of $p$-valued groups played an instrumental role in the study of locally analytic representations. In a previous work we attached a rigid analytic group to a $\textit{$p$-saturated group}$ (a class of $p$-valued groups that contains uniform pro-$p$ groups and pro-$p$ Iwahori subgroups as examples). In this talk I will outline how to elevate the rigid group to a $\textit{dagger group}$, a group object in the category of dagger spaces as introduced by Elmar Grosse-Klönne. I will further introduce the space of $\textit{overconvergent functions}$ and its strong dual the $\textit{overconvergent distribution algebra}$ on such a group. Finally I will show that in analogy to the locally analytic distribution algebra of compact $p$-adic groups, in the case of uniform pro-$p$ groups the overconvergent distribution algebra is a Fr´echet-Stein algebra, i.e., it is equipped with a desirable algebraic structure. This is joint work with Claus Sorensen and Matthias Strauch.

### February 29, 2024

### 2:00 PM

APM 7321

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